Subgroups — C₆₁×#23809 #29527242

Order 15616.

974 conjugacy classes of subgroups.

The subgroup lattice has 604 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₆₁×#441	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₆₁×#440	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×SD₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#2327	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₁×#2326	C₂	1	1	yes	maximal, characteristic
7	C₆₁×#2097	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₁×#2096	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₁×#2095	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₁×#2092	C₂	1	1	yes	maximal, characteristic
11	C₆₁×#2016	C₂	1	1	yes	maximal, characteristic, complemented
12	C₆₁×#2015	C₂	1	1	yes	maximal, characteristic
13	C₆₁×#2015	C₂	1	1	yes	maximal, characteristic
14	C₆₁×#2013	C₂	1	1	yes	maximal, characteristic
15	C₆₁×#1746	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
18	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×SD₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₂×#124×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₆₁	C₂²	1	1	yes	characteristic
26	C₆₁×#441	—	2	4	no	—
27	C₆₁×#440	—	2	4	no	—
28	C₄×Q₁₆×C₆₁	—	2	4	no	—
29	C₄×SD₁₆×C₆₁	—	2	4	no	—
30	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
31	C₂×#107×C₆₁	C₂²	1	1	yes	characteristic
32	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
33	C₂×#106×C₆₁	C₂²	1	1	yes	characteristic
34	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
35	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
36	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
37	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
38	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
39	C₂×#103×C₆₁	C₂²	1	1	yes	characteristic
40	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
41	C₂×#102×C₆₁	C₂²	1	1	yes	characteristic
42	C₂×#97×C₆₁	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
44	C₆₁×#391	C₂²	1	1	yes	characteristic
45	C₆₁×#391	C₂²	1	1	yes	characteristic
46	C₆₁×#391	C₂²	1	1	yes	characteristic
47	C₆₁×#390	C₂²	1	1	yes	characteristic
48	C₆₁×#388	—	2	2	no	—
49	C₆₁×#387	—	2	2	no	—
50	C₆₁×#387	—	2	2	no	—
51	C₆₁×#385	—	2	2	no	—
52	C₆₁×#384	C₂²	1	1	yes	characteristic
53	C₆₁×#382	C₂²	1	1	yes	characteristic
54	C₆₁×#376	C₂²	1	1	yes	characteristic
55	C₆₁×#336	C₂²	1	1	yes	characteristic
56	C₆₁×#336	C₂²	1	1	yes	characteristic
57	C₆₁×#336	C₂²	1	1	yes	characteristic
58	C₆₁×#336	C₂²	1	1	yes	characteristic
59	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
60	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
61	C₂²×Q₈×C₆₁	C₂³	1	1	yes	characteristic
62	C₂²×D₄×C₆₁	C₂³	1	1	yes	characteristic
63	C₂³×C₄×C₆₁	—	2	1	no	—
64	C₂×Q₁₆×C₆₁	—	2	2	no	—
65	C₂×Q₁₆×C₆₁	—	2	2	no	—
66	C₂×Q₁₆×C₆₁	—	2	2	no	—
67	C₂×Q₁₆×C₆₁	—	2	2	no	—
68	C₂×Q₁₆×C₆₁	—	2	2	no	—
69	C₂×Q₁₆×C₆₁	—	2	2	no	—
70	C₂×SD₁₆×C₆₁	—	2	2	no	—
71	C₂×SD₁₆×C₆₁	—	2	2	no	—
72	C₂×SD₁₆×C₆₁	—	2	2	no	—
73	C₂×SD₁₆×C₆₁	—	2	2	no	—
74	C₂×SD₁₆×C₆₁	—	2	2	no	—
75	C₂×SD₁₆×C₆₁	—	2	2	no	—
76	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
78	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
79	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
80	C₄⋊Q₈×C₆₁	—	2	2	no	—
81	C₄⋊Q₈×C₆₁	—	2	2	no	—
82	#124×C₆₁	—	2	2	no	—
83	#124×C₆₁	—	2	2	no	—
84	C₄×Q₈×C₆₁	—	2	2	no	—
85	C₄×Q₈×C₆₁	—	2	2	no	—
86	C₄×Q₈×C₆₁	—	2	2	no	—
87	C₄×Q₈×C₆₁	—	2	2	no	—
88	C₄×Q₈×C₆₁	—	2	2	no	—
89	C₄×Q₈×C₆₁	—	2	2	no	—
90	C₄×D₄×C₆₁	—	2	2	no	—
91	C₄×D₄×C₆₁	—	2	2	no	—
92	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
93	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
94	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
95	C₂×C₄⋊C₄×C₆₁	—	2	1	no	—
96	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
97	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₆₁	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
102	C₂×C₂²⋊C₄×C₆₁	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₆₁	—	2	2	no	—
104	C₂×C₄²×C₆₁	—	2	1	no	—
105	C₂×C₄²×C₆₁	—	2	1	no	—
106	C₂×C₄²×C₆₁	—	2	1	no	—
107	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
108	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
109	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
110	#107×C₆₁	—	2	2	no	—
111	#107×C₆₁	—	2	2	no	—
112	#106×C₆₁	—	2	2	no	—
113	#106×C₆₁	—	2	2	no	—
114	#103×C₆₁	—	2	2	no	—
115	#103×C₆₁	—	2	2	no	—
116	#103×C₆₁	—	2	2	no	—
117	#103×C₆₁	—	2	2	no	—
118	#103×C₆₁	—	2	2	no	—
119	#103×C₆₁	—	2	2	no	—
120	#102×C₆₁	—	2	2	no	—
121	#102×C₆₁	—	2	2	no	—
122	#97×C₆₁	D₄	1	4	yes	—
123	C₄×C₈×C₆₁	D₄	1	4	yes	—
124	#95×C₆₁	—	2	2	no	—
125	#95×C₆₁	—	2	2	no	—
126	#95×C₆₁	—	2	2	no	—
127	#95×C₆₁	—	2	2	no	—
128	#95×C₆₁	D₄	1	2	yes	—
129	#95×C₆₁	D₄	1	2	yes	—
130	C₂⁴×C₆₁	—	2	1	no	—
131	C₂×Q₈×C₆₁	—	2	1	no	—
132	C₂×Q₈×C₆₁	—	2	1	no	—
133	C₂×Q₈×C₆₁	—	2	1	no	—
134	C₂×Q₈×C₆₁	—	2	1	no	—
135	C₂×Q₈×C₆₁	—	2	1	no	—
136	C₂×Q₈×C₆₁	—	2	1	no	—
137	C₂×Q₈×C₆₁	—	2	1	no	—
138	C₂×Q₈×C₆₁	—	2	1	no	—
139	C₂×Q₈×C₆₁	—	2	1	no	—
140	C₂×Q₈×C₆₁	—	2	1	no	—
141	C₂×Q₈×C₆₁	—	2	1	no	—
142	C₂×Q₈×C₆₁	—	2	1	no	—
143	C₂×Q₈×C₆₁	—	2	1	no	—
144	C₂×Q₈×C₆₁	—	2	1	no	—
145	C₂×Q₈×C₆₁	—	2	1	no	—
146	C₂×Q₈×C₆₁	#41	1	2	yes	—
147	C₂×Q₈×C₆₁	#41	1	2	yes	—
148	C₂×Q₈×C₆₁	#41	1	2	yes	—
149	C₂×D₄×C₆₁	—	2	1	no	—
150	C₂×D₄×C₆₁	—	2	1	no	—
151	C₂×D₄×C₆₁	—	2	1	no	—
152	C₂×D₄×C₆₁	—	2	1	no	—
153	C₂×D₄×C₆₁	—	2	1	no	—
154	C₂×D₄×C₆₁	#41	1	2	yes	—
155	C₂²×C₄×C₆₁	—	2	1	no	—
156	C₂²×C₄×C₆₁	—	2	1	no	—
157	C₂²×C₄×C₆₁	—	2	1	no	—
158	C₂²×C₄×C₆₁	—	2	1	no	—
159	C₂²×C₄×C₆₁	—	2	1	no	—
160	C₂²×C₄×C₆₁	—	2	1	no	—
161	C₂²×C₄×C₆₁	—	2	1	no	—
162	C₂²×C₄×C₆₁	—	4	2	no	—
163	C₂²×C₄×C₆₁	—	8	2	no	—
164	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₆₁	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
169	Q₁₆×C₆₁	—	2	8	no	—
170	Q₁₆×C₆₁	—	2	8	no	—
171	SD₁₆×C₆₁	—	2	8	no	—
172	SD₁₆×C₆₁	—	2	8	no	—
173	C₂×C₈×C₆₁	—	2	1	no	—
174	C₂×C₈×C₆₁	—	2	1	no	—
175	C₂×C₈×C₆₁	—	2	1	no	—
176	C₂×C₈×C₆₁	—	2	1	no	—
177	C₂×C₈×C₆₁	—	2	1	no	—
178	C₂×C₈×C₆₁	—	2	1	no	—
179	C₂×C₈×C₆₁	—	2	1	no	—
180	C₂×C₈×C₆₁	—	2	1	no	—
181	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
182	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
183	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
184	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
185	C₄⋊C₄×C₆₁	—	2	8	no	—
186	C₄⋊C₄×C₆₁	—	2	1	no	—
187	C₄⋊C₄×C₆₁	—	2	1	no	—
188	C₄⋊C₄×C₆₁	—	2	1	no	—
189	C₄⋊C₄×C₆₁	—	2	1	no	—
190	C₄⋊C₄×C₆₁	—	2	1	no	—
191	C₄⋊C₄×C₆₁	—	2	1	no	—
192	C₄⋊C₄×C₆₁	—	2	1	no	—
193	C₄⋊C₄×C₆₁	—	2	1	no	—
194	C₄⋊C₄×C₆₁	—	4	2	no	—
195	C₄⋊C₄×C₆₁	—	4	2	no	—
196	C₄⋊C₄×C₆₁	—	4	2	no	—
197	C₄⋊C₄×C₆₁	—	8	2	no	—
198	C₂²⋊C₄×C₆₁	—	2	8	no	—
199	C₂²⋊C₄×C₆₁	—	4	2	no	—
200	C₂²⋊C₄×C₆₁	—	8	2	no	—
201	C₄²×C₆₁	—	2	1	no	—
202	C₄²×C₆₁	—	2	1	no	—
203	C₄²×C₆₁	—	2	1	no	—
204	C₄²×C₆₁	—	2	1	no	—
205	C₄²×C₆₁	—	4	2	no	—
206	C₄²×C₆₁	—	4	2	no	—
207	C₄²×C₆₁	—	4	2	no	—
208	C₄²×C₆₁	C₂×D₄	1	2	yes	—
209	C₄²×C₆₁	C₂×D₄	1	2	yes	—
210	C₂³×C₆₁	—	2	2	no	—
211	C₂³×C₆₁	—	4	1	no	—
212	C₂³×C₆₁	—	4	1	no	—
213	C₂³×C₆₁	—	4	2	no	—
214	C₂³×C₆₁	—	4	2	no	—
215	C₂³×C₆₁	C₂²×D₄	1	1	yes	characteristic
216	Q₈×C₆₁	—	2	2	no	—
217	Q₈×C₆₁	—	2	2	no	—
218	Q₈×C₆₁	—	2	2	no	—
219	Q₈×C₆₁	—	2	2	no	—
220	Q₈×C₆₁	—	2	2	no	—
221	Q₈×C₆₁	—	2	2	no	—
222	Q₈×C₆₁	—	4	1	no	—
223	Q₈×C₆₁	—	4	1	no	—
224	Q₈×C₆₁	—	4	1	no	—
225	Q₈×C₆₁	—	4	1	no	—
226	Q₈×C₆₁	—	4	1	no	—
227	Q₈×C₆₁	—	4	1	no	—
228	D₄×C₆₁	—	2	2	no	—
229	D₄×C₆₁	—	2	2	no	—
230	D₄×C₆₁	—	4	1	no	—
231	D₄×C₆₁	—	4	1	no	—
232	C₂×C₄×C₆₁	—	2	2	no	—
233	C₂×C₄×C₆₁	—	2	2	no	—
234	C₂×C₄×C₆₁	—	2	2	no	—
235	C₂×C₄×C₆₁	—	2	2	no	—
236	C₂×C₄×C₆₁	—	2	2	no	—
237	C₂×C₄×C₆₁	—	2	2	no	—
238	C₂×C₄×C₆₁	—	2	2	no	—
239	C₂×C₄×C₆₁	—	2	2	no	—
240	C₂×C₄×C₆₁	—	2	2	no	—
241	C₂×C₄×C₆₁	—	2	2	no	—
242	C₂×C₄×C₆₁	—	2	2	no	—
243	C₂×C₄×C₆₁	—	4	1	no	—
244	C₂×C₄×C₆₁	—	4	1	no	—
245	C₂×C₄×C₆₁	—	4	1	no	—
246	C₂×C₄×C₆₁	—	4	1	no	—
247	C₂×C₄×C₆₁	—	4	1	no	—
248	C₂×C₄×C₆₁	—	4	1	no	—
249	C₂×C₄×C₆₁	—	4	1	no	—
250	C₂×C₄×C₆₁	—	4	1	no	—
251	C₂×C₄×C₆₁	—	4	1	no	—
252	C₂×C₄×C₆₁	—	4	1	no	—
253	C₂×C₄×C₆₁	—	4	2	no	—
254	C₂×C₄×C₆₁	—	4	2	no	—
255	C₂×C₄×C₆₁	—	4	2	no	—
256	C₂×C₄×C₆₁	—	4	2	no	—
257	C₂×C₄×C₆₁	—	8	4	no	—
258	C₂×C₄×C₆₁	#127	1	2	yes	—
259	C₂×C₄×C₆₁	#127	1	2	yes	—
260	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
261	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
262	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
263	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
264	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
265	C₂×C₄×C₆₁	C₂×#41	1	1	yes	characteristic
266	C₈×C₆₁	—	2	2	no	—
267	C₈×C₆₁	—	2	2	no	—
268	C₈×C₆₁	—	2	2	no	—
269	C₈×C₆₁	—	2	2	no	—
270	#23809	C₆₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
271	C₂²×C₆₁	—	4	4	no	—
272	C₂²×C₆₁	—	4	2	no	—
273	C₂²×C₆₁	—	8	2	no	—
274	C₂²×C₆₁	—	8	1	no	—
275	C₂²×C₆₁	—	8	1	no	—
276	C₂²×C₆₁	#525	1	1	yes	characteristic
277	C₂²×C₆₁	#525	1	1	yes	characteristic
278	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
279	C₂²×C₆₁	#577	1	2	yes	—
280	C₂²×C₆₁	#578	1	2	yes	—
281	C₄×C₆₁	—	2	2	no	—
282	C₄×C₆₁	—	2	2	no	—
283	C₄×C₆₁	—	4	2	no	—
284	C₄×C₆₁	—	4	2	no	—
285	C₄×C₆₁	—	4	2	no	—
286	C₄×C₆₁	—	4	2	no	—
287	C₄×C₆₁	—	4	2	no	—
288	C₄×C₆₁	—	8	1	no	—
289	C₄×C₆₁	—	8	1	no	—
290	C₄×C₆₁	—	8	1	no	—
291	C₄×C₆₁	—	8	1	no	—
292	C₄×C₆₁	#525	1	1	yes	characteristic
293	C₄×C₆₁	#525	1	1	yes	characteristic
294	C₄×C₆₁	#525	1	1	yes	characteristic
295	C₄×C₆₁	#525	1	1	yes	characteristic
296	C₂×#441	C₂×C₆₁	1	1	yes	characteristic, complemented
297	C₂×#440	C₂×C₆₁	1	1	yes	characteristic
298	C₂×C₄×Q₁₆	C₂×C₆₁	1	1	yes	characteristic, complemented
299	C₂×C₄×SD₁₆	C₂×C₆₁	1	1	yes	characteristic
300	#2327	C₂×C₆₁	1	1	yes	characteristic, complemented
301	#2326	C₂×C₆₁	1	1	yes	characteristic
302	#2097	C₂×C₆₁	1	1	yes	characteristic, complemented
303	#2096	C₂×C₆₁	1	1	yes	characteristic, complemented
304	#2095	C₂×C₆₁	1	1	yes	characteristic, complemented
305	#2092	C₂×C₆₁	1	1	yes	characteristic
306	#2016	C₂×C₆₁	1	1	yes	characteristic, complemented
307	#2015	C₂×C₆₁	1	1	yes	characteristic
308	#2015	C₂×C₆₁	1	1	yes	characteristic
309	#2013	C₂×C₆₁	1	1	yes	characteristic
310	#1746	C₂×C₆₁	1	1	yes	characteristic, complemented
311	C₂×C₆₁	—	8	2	no	—
312	C₂×C₆₁	#2397	1	1	yes	characteristic
313	C₂×C₆₁	#3120	1	2	yes	—
314	C₂×C₆₁	#3121	1	2	yes	—
315	C₂×C₆₁	#3154	1	1	yes	characteristic
316	C₂×C₆₁	#3156	1	1	yes	characteristic
317	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
318	C₂²×Q₁₆	C₂²×C₆₁	1	1	yes	characteristic
319	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
320	C₂²×SD₁₆	C₂²×C₆₁	1	1	yes	characteristic
321	C₂×C₄⋊Q₈	C₂²×C₆₁	1	1	yes	characteristic
322	C₂×#124	C₂²×C₆₁	1	1	yes	characteristic
323	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
324	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
325	C₂×C₄×Q₈	C₂²×C₆₁	1	1	yes	characteristic
326	C₂×C₄×D₄	C₂²×C₆₁	1	1	yes	characteristic
327	#441	—	2	4	no	—
328	#440	—	2	4	no	—
329	C₄×Q₁₆	—	2	4	no	—
330	C₄×SD₁₆	—	2	4	no	—
331	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
332	C₂×#107	C₂²×C₆₁	1	1	yes	characteristic
333	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
334	C₂×#106	C₂²×C₆₁	1	1	yes	characteristic
335	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
336	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
337	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
338	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
339	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
340	C₂×#103	C₂²×C₆₁	1	1	yes	characteristic
341	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
342	C₂×#102	C₂²×C₆₁	1	1	yes	characteristic
343	C₂×#97	C₂²×C₆₁	1	1	yes	characteristic
344	C₂×C₄×C₈	C₂²×C₆₁	1	1	yes	characteristic
345	#391	C₂²×C₆₁	1	1	yes	characteristic
346	#391	C₂²×C₆₁	1	1	yes	characteristic
347	#391	C₂²×C₆₁	1	1	yes	characteristic
348	#390	C₂²×C₆₁	1	1	yes	characteristic
349	#388	—	2	2	no	—
350	#387	—	2	2	no	—
351	#387	—	2	2	no	—
352	#385	—	2	2	no	—
353	#384	C₂²×C₆₁	1	1	yes	characteristic
354	#382	C₂²×C₆₁	1	1	yes	characteristic
355	#376	C₂²×C₆₁	1	1	yes	characteristic
356	#336	C₂²×C₆₁	1	1	yes	characteristic
357	#336	C₂²×C₆₁	1	1	yes	characteristic
358	#336	C₂²×C₆₁	1	1	yes	characteristic
359	#336	C₂²×C₆₁	1	1	yes	characteristic
360	C₆₁	#23809	1	1	yes	characteristic, direct product factor, Sylow
361	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
362	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
363	C₂²×Q₈	C₂³×C₆₁	1	1	yes	characteristic
364	C₂²×D₄	C₂³×C₆₁	1	1	yes	characteristic
365	C₂³×C₄	—	2	1	no	—
366	C₂×Q₁₆	—	2	2	no	—
367	C₂×Q₁₆	—	2	2	no	—
368	C₂×Q₁₆	—	2	2	no	—
369	C₂×Q₁₆	—	2	2	no	—
370	C₂×Q₁₆	—	2	2	no	—
371	C₂×Q₁₆	—	2	2	no	—
372	C₂×SD₁₆	—	2	2	no	—
373	C₂×SD₁₆	—	2	2	no	—
374	C₂×SD₁₆	—	2	2	no	—
375	C₂×SD₁₆	—	2	2	no	—
376	C₂×SD₁₆	—	2	2	no	—
377	C₂×SD₁₆	—	2	2	no	—
378	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
379	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
380	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
381	C₂²×C₈	C₂³×C₆₁	1	1	yes	characteristic
382	C₄⋊Q₈	—	2	2	no	—
383	C₄⋊Q₈	—	2	2	no	—
384	#124	—	2	2	no	—
385	#124	—	2	2	no	—
386	C₄×Q₈	—	2	2	no	—
387	C₄×Q₈	—	2	2	no	—
388	C₄×Q₈	—	2	2	no	—
389	C₄×Q₈	—	2	2	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×D₄	—	2	2	no	—
393	C₄×D₄	—	2	2	no	—
394	C₂×C₄⋊C₄	—	2	2	no	—
395	C₂×C₄⋊C₄	—	2	1	no	—
396	C₂×C₄⋊C₄	—	2	1	no	—
397	C₂×C₄⋊C₄	—	2	1	no	—
398	C₂×C₄⋊C₄	—	2	2	no	—
399	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
400	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
401	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
402	C₂×C₄⋊C₄	C₂³×C₆₁	1	1	yes	characteristic
403	C₂×C₂²⋊C₄	—	2	2	no	—
404	C₂×C₂²⋊C₄	—	2	1	no	—
405	C₂×C₂²⋊C₄	—	2	2	no	—
406	C₂×C₄²	—	2	1	no	—
407	C₂×C₄²	—	2	1	no	—
408	C₂×C₄²	—	2	1	no	—
409	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
410	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
411	C₂×C₄²	C₂³×C₆₁	1	1	yes	characteristic
412	#107	—	2	2	no	—
413	#107	—	2	2	no	—
414	#106	—	2	2	no	—
415	#106	—	2	2	no	—
416	#103	—	2	2	no	—
417	#103	—	2	2	no	—
418	#103	—	2	2	no	—
419	#103	—	2	2	no	—
420	#103	—	2	2	no	—
421	#103	—	2	2	no	—
422	#102	—	2	2	no	—
423	#102	—	2	2	no	—
424	#97	D₄×C₆₁	1	4	yes	—
425	C₄×C₈	D₄×C₆₁	1	4	yes	—
426	#95	—	2	2	no	—
427	#95	—	2	2	no	—
428	#95	—	2	2	no	—
429	#95	—	2	2	no	—
430	#95	D₄×C₆₁	1	2	yes	—
431	#95	D₄×C₆₁	1	2	yes	—
432	C₂⁴	—	2	1	no	—
433	C₂×Q₈	—	2	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	1	no	—
436	C₂×Q₈	—	2	1	no	—
437	C₂×Q₈	—	2	1	no	—
438	C₂×Q₈	—	2	1	no	—
439	C₂×Q₈	—	2	1	no	—
440	C₂×Q₈	—	2	1	no	—
441	C₂×Q₈	—	2	1	no	—
442	C₂×Q₈	—	2	1	no	—
443	C₂×Q₈	—	2	1	no	—
444	C₂×Q₈	—	2	1	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	1	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	#41×C₆₁	1	2	yes	—
449	C₂×Q₈	#41×C₆₁	1	2	yes	—
450	C₂×Q₈	#41×C₆₁	1	2	yes	—
451	C₂×D₄	—	2	1	no	—
452	C₂×D₄	—	2	1	no	—
453	C₂×D₄	—	2	1	no	—
454	C₂×D₄	—	2	1	no	—
455	C₂×D₄	—	2	1	no	—
456	C₂×D₄	#41×C₆₁	1	2	yes	—
457	C₂²×C₄	—	2	1	no	—
458	C₂²×C₄	—	2	1	no	—
459	C₂²×C₄	—	2	1	no	—
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	1	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	4	2	no	—
465	C₂²×C₄	—	8	2	no	—
466	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
467	C₂²×C₄	C₂×D₄×C₆₁	1	2	yes	—
468	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
469	C₂²×C₄	C₂×D₄×C₆₁	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₆₁	1	1	yes	characteristic
471	Q₁₆	—	2	8	no	—
472	Q₁₆	—	2	8	no	—
473	SD₁₆	—	2	8	no	—
474	SD₁₆	—	2	8	no	—
475	C₂×C₈	—	2	1	no	—
476	C₂×C₈	—	2	1	no	—
477	C₂×C₈	—	2	1	no	—
478	C₂×C₈	—	2	1	no	—
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	1	no	—
482	C₂×C₈	—	2	1	no	—
483	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
484	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
485	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
486	C₂×C₈	C₂×D₄×C₆₁	1	2	yes	—
487	C₄⋊C₄	—	2	8	no	—
488	C₄⋊C₄	—	2	1	no	—
489	C₄⋊C₄	—	2	1	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	1	no	—
494	C₄⋊C₄	—	2	1	no	—
495	C₄⋊C₄	—	2	1	no	—
496	C₄⋊C₄	—	4	2	no	—
497	C₄⋊C₄	—	4	2	no	—
498	C₄⋊C₄	—	4	2	no	—
499	C₄⋊C₄	—	8	2	no	—
500	C₂²⋊C₄	—	2	8	no	—
501	C₂²⋊C₄	—	4	2	no	—
502	C₂²⋊C₄	—	8	2	no	—
503	C₄²	—	2	1	no	—
504	C₄²	—	2	1	no	—
505	C₄²	—	2	1	no	—
506	C₄²	—	2	1	no	—
507	C₄²	—	4	2	no	—
508	C₄²	—	4	2	no	—
509	C₄²	—	4	2	no	—
510	C₄²	C₂×D₄×C₆₁	1	2	yes	—
511	C₄²	C₂×D₄×C₆₁	1	2	yes	—
512	C₂³	—	2	2	no	—
513	C₂³	—	4	1	no	—
514	C₂³	—	4	2	no	—
515	C₂³	—	4	2	no	—
516	C₂³	—	4	1	no	—
517	C₂³	C₂²×D₄×C₆₁	1	1	yes	characteristic
518	Q₈	—	2	2	no	—
519	Q₈	—	2	2	no	—
520	Q₈	—	2	2	no	—
521	Q₈	—	2	2	no	—
522	Q₈	—	2	2	no	—
523	Q₈	—	2	2	no	—
524	Q₈	—	4	1	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	4	1	no	—
527	Q₈	—	4	1	no	—
528	Q₈	—	4	1	no	—
529	Q₈	—	4	1	no	—
530	D₄	—	2	2	no	—
531	D₄	—	2	2	no	—
532	D₄	—	4	1	no	—
533	D₄	—	4	1	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	2	2	no	—
537	C₂×C₄	—	2	2	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	2	no	—
541	C₂×C₄	—	2	2	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	4	1	no	—
546	C₂×C₄	—	4	1	no	—
547	C₂×C₄	—	4	1	no	—
548	C₂×C₄	—	4	1	no	—
549	C₂×C₄	—	4	1	no	—
550	C₂×C₄	—	4	1	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	1	no	—
554	C₂×C₄	—	4	1	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	2	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	8	4	no	—
560	C₂×C₄	#127×C₆₁	1	2	yes	—
561	C₂×C₄	#127×C₆₁	1	2	yes	—
562	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
563	C₂×C₄	C₂²×D₄×C₆₁	1	1	yes	characteristic
564	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
565	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
566	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
567	C₂×C₄	C₂×#41×C₆₁	1	1	yes	characteristic
568	C₈	—	2	2	no	—
569	C₈	—	2	2	no	—
570	C₈	—	2	2	no	—
571	C₈	—	2	2	no	—
572	C₂²	—	4	4	no	—
573	C₂²	—	4	2	no	—
574	C₂²	—	8	2	no	—
575	C₂²	—	8	1	no	—
576	C₂²	—	8	1	no	—
577	C₂²	C₆₁×#525	1	1	yes	characteristic
578	C₂²	C₆₁×#525	1	1	yes	characteristic
579	C₂²	C₂×#127×C₆₁	1	1	yes	characteristic
580	C₂²	C₆₁×#577	1	2	yes	—
581	C₂²	C₆₁×#578	1	2	yes	—
582	C₄	—	2	2	no	—
583	C₄	—	2	2	no	—
584	C₄	—	4	2	no	—
585	C₄	—	4	2	no	—
586	C₄	—	4	2	no	—
587	C₄	—	4	2	no	—
588	C₄	—	4	2	no	—
589	C₄	—	8	1	no	—
590	C₄	—	8	1	no	—
591	C₄	—	8	1	no	—
592	C₄	—	8	1	no	—
593	C₄	C₆₁×#525	1	1	yes	characteristic
594	C₄	C₆₁×#525	1	1	yes	characteristic
595	C₄	C₆₁×#525	1	1	yes	characteristic
596	C₄	C₆₁×#525	1	1	yes	characteristic
597	C₂	—	8	2	no	—
598	C₂	C₆₁×#2397	1	1	yes	characteristic
599	C₂	C₆₁×#3120	1	2	yes	—
600	C₂	C₆₁×#3121	1	2	yes	—
601	C₂	C₆₁×#3154	1	1	yes	characteristic
602	C₂	C₆₁×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₁	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₁
prime radical	C₂²×D₄×C₆₁	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₁ ◁ C₆₁×#376 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₆₁×#441	C₂	semidirect
3. C₂×C₄×Q₁₆×C₆₁	C₂	semidirect
5. C₆₁×#2327	C₂	semidirect
7. C₆₁×#2097	C₂	semidirect
8. C₆₁×#2096	C₂	semidirect
9. C₆₁×#2095	C₂	semidirect
11. C₆₁×#2016	C₂	semidirect
15. C₆₁×#1746	C₂	semidirect
270. #23809	C₆₁	direct (N × H)
296. C₂×#441	C₂×C₆₁	semidirect
298. C₂×C₄×Q₁₆	C₂×C₆₁	semidirect
300. #2327	C₂×C₆₁	semidirect
302. #2097	C₂×C₆₁	semidirect
303. #2096	C₂×C₆₁	semidirect
304. #2095	C₂×C₆₁	semidirect
306. #2016	C₂×C₆₁	semidirect
310. #1746	C₂×C₆₁	semidirect
360. C₆₁	#23809	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₆₁×#441	2
C₂×C₆₁×#440	2
C₂×C₄×Q₁₆×C₆₁	2
C₂×C₄×SD₁₆×C₆₁	2
C₆₁×#2327	2
C₆₁×#2326	2
C₆₁×#2097	2
C₆₁×#2096	2
C₆₁×#2095	2
C₆₁×#2092	2
C₆₁×#2016	2
C₆₁×#2015	2
C₆₁×#2013	2
C₆₁×#1746	2
#23809	61

Subgroups — C61×#23809 #29527242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₁×#23809 #29527242